数学代写|常微分方程代写ordinary differential equation代考|MATH3331

2023年2月2日

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数学代写|常微分方程代写ordinary differential equation代考|Convergence of the Picard Sequence

Let us now look more closely at the Picard sequence of functions,
$$\psi_0, \psi_1, \psi_2, \psi_3, \ldots$$
with $\psi_0$ being “some continuous function” and
$$\psi_{k+1}(x)=y_0+\int_0^x F\left(s, \psi_k(s)\right) d s \quad \text { for } \quad k=0,1,2,3, \ldots .$$
Remember, $F$ and ${ }^{\partial F} / \partial y$ are continuous on some open region containing the point $\left(0, y_0\right)$. This means Lemma $3.5$ applies. Let $[\alpha, \beta], M, B$ and $\Delta Y$ be the interval and constants from that lemma. Let us also now impose an additional restriction on the choice for $\psi_0$ : Let us insist that $\psi_0$ be any continuous function on $[\alpha, \beta]$ such that
$$\left|\psi_0(x)-y_0\right| \leq \Delta Y \quad \text { for } \quad \alpha<x<\beta .$$
In particular, we could let $\psi_0$ be the constant function $\psi_0(x)=y_0$ for all $x$.
We now want to show that the sequence of $\psi_k$ ‘s converges to a function $y$ on $[\alpha, \beta]$. Our first step in this direction is to observe that, thanks to the additional requirement on $\psi_0$, Lemma $3.5$ can be applied repeatedly to show that $\psi_1, \psi_2, \psi_3, \ldots$ are all well-defined, continuous functions on the interval $[\alpha, \beta]$ with each satisfying
$$\left|\psi_k(x)-y_0\right| \leq \Delta Y \quad \text { for } \quad \alpha \leq x \leq \beta .$$
Next, we need to establish useful bounds on the sequence
$$\left|\psi_1(x)-\psi_0(x)\right| \quad, \quad\left|\psi_2(x)-\psi_1(x)\right| \quad, \quad\left|\psi_3(x)-\psi_2(x)\right| \quad, \quad \ldots$$
when $\alpha \leq x \leq \beta$. The first is easy:
\begin{aligned} \left|\psi_1(x)-\psi_0(x)\right| & =\left|\psi_1(x)-y_0-\psi_0(x)+y_0\right| \ & =\left|\left[\psi_1(x)-y_0\right]+\left(-\left[\psi_0(x)-y_0\right]\right)\right| \ & \leq\left|\psi_1(x)-y_0\right|+\left|\psi_0(x)-y_0\right| \leq 2 \Delta Y \end{aligned}

数学代写|常微分方程代写ordinary differential equation代考|The Uniqueness Claim in Theorem

If you’ve made it through this section up to this point, then you should have little difficulty in finishing the proof of Theorem $3.1$ by doing the following exercises. Do make use of the work we’ve done in the previous several pages.
?-Exercise 3.2: Consider a first-order initial-value problem
$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y(0)=y_0,$$ and with both $F$ and ${ }^{\partial F / \partial y}$ being continuous functions on some open region containing the point $\left(0, y_0\right)$. Since Lemma $3.5$ applies, we can let $[\alpha, \beta]$ be the interval, and $M, B$ and $\Delta Y$ the positive constants from that lemma. Using this interval and these constants:
a i: Verify that
$$0 \leq M|x| \leq \Delta Y \quad \text { for } \quad \alpha \leq x \leq \beta .$$
ii: Also verify that any solution $y$ to the above initial-value problem satisfies
$$\left|y(x)-y_0\right| \leq M|x| \quad \text { for } a<x<b .$$
Now observe that the last two inequalities yield
$$\left|y(x)-y_0\right| \leq M|x| \leq \Delta Y \quad \text { for } \quad \alpha \leq x \leq \beta$$
whenever $y$ is a solution to the above initial-value problem.
b: For the following, let $y_1$ and $y_2$ be any two solutions to the above initial-value problem on $(\alpha, \beta)$, and let
$$\psi_0, \psi_1, \psi_2, \psi_3, \ldots \quad \text { and } \phi_0, \phi_1, \phi_2, \phi_3, \ldots$$
be the two Picard sequences of functions on $(\alpha, \beta)$ generated by setting
and
$$\psi_{k+1}(x)=y_0+\int_0^x F\left(s, \psi_k(s)\right) d s$$
$$\phi_{k+1}(x)=y_0+\int_0^x F\left(s, \phi_k(s)\right) d s$$
with
$$\psi_0(x)=y_1(x) \quad \text { and } \quad \phi_0(x)=y_2(x)$$

常微分方程代写

数学代写|常微分方程代写ordinary differential equation代考|Convergence of the Picard Sequence

$$\psi_0, \psi_1, \psi_2, \psi_3, \ldots$$

$$\psi_{k+1}(x)=y_0+\int_0^x F\left(s, \psi_k(s)\right) d s \quad \text { for } \quad k=0,1 \text {, }$$

$$\left|\psi_0(x)-y_0\right| \leq \Delta Y \quad \text { for } \quad \alpha<x<\beta .$$

$$\left|\psi_k(x)-y_0\right| \leq \Delta Y \quad \text { for } \quad \alpha \leq x \leq \beta$$

$$\left|\psi_1(x)-\psi_0(x)\right| \quad, \quad\left|\psi_2(x)-\psi_1(x)\right| \quad, \quad \mid \psi_3(x)$$

$$\left|\psi_1(x)-\psi_0(x)\right|=\left|\psi_1(x)-y_0-\psi_0(x)+y_0\right|$$

数学代写|常微分方程代写ordinary differential equation代考|The Uniqueness Claim in Theorem

?-练习 3.2：考虑一阶初值问题
$$\frac{d y}{d x}=F(x, y) \quad \text { with } \quad y(0)=y_0,$$

ai: 验证
$$0 \leq M|x| \leq \Delta Y \quad \text { for } \quad \alpha \leq x \leq \beta$$
ii：还要验证任何解决方案 $y$ 对上述初值问题满足
$$\left|y(x)-y_0\right| \leq M|x| \quad \text { for } a<x<b .$$

$$\left|y(x)-y_0\right| \leq M|x| \leq \Delta Y \quad \text { for } \quad \alpha \leq x \leq \beta$$

$\mathrm{b}$ : 对于以下，让 $y_1$ 和 $y_2$ 是上述初值问题的任意两个解 $(\alpha, \beta)$ ，然后让
$\psi_0, \psi_1, \psi_2, \psi_3, \ldots \quad$ and $\phi_0, \phi_1, \phi_2, \phi_3, \ldots$

$$\psi_{k+1}(x)=y_0+\int_0^x F\left(s, \psi_k(s)\right) d s$$
$$\phi_{k+1}(x)=y_0+\int_0^x F\left(s, \phi_k(s)\right) d s$$

$\psi_0(x)=y_1(x) \quad$ and $\quad \phi_0(x)=y_2(x)$

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